In this paper, we introduce a notion of generalized potential games that is inspired by a newly developed theory on generalized gradient flows. More precisely, a game is called generalized potential if the simultaneous gradient of the loss functions is a nonlinear function of the gradient of a potential function. Applications include a class of games arising from chemical reaction networks with detailed balance condition. For this class of games, we prove an explicit exponential convergence to equilibrium for evolution of a single reversible reaction. Moreover, numerical investigations are performed to calculate the equilibrium state of some reversible chemical reactions which give rise to generalized potential games.
It is known that the generalized Nash equilibrium problem can be reformulated as a quasivariational inequality. Our aim in this work is to introduce a variational approach to study the existence of solutions for generalized ordinal Nash games, that is, generalized games where the player preferences are binary relations that do not necessarily admit a utility representation.
In this paper we study the two-player generalized Cops and Robber (GCR) games introduced by Bonato and MacGillivray. Our main goal is to present a full, self-contained game theoretic analysis of such games.
In the Colonel Blotto game, two players with a fixed budget simultaneously allocate their resources across n battlefields to maximize the aggregate value gained from the battlefields where they have the higher allocation. Despite its long-standing history and important applications, the Colonel Blotto game still lacks a complete Nash equilibrium characterization in its most general form where players are asymmetric and battlefields' values are heterogeneous across battlefields and different between the two players---this is called the Generalized Colonel Blotto game. In this work, we propose a…
In this paper, we show that different types of evolutionary game dynamics are, in principle, special cases of a dynamical system model based on our previously reported framework of generalized growth transforms. The framework shows that different dynamics arise as a result of minimizing a population energy such that the population as a whole evolves to reach the most stable state. By introducing a population dependent time-constant in the generalized growth transform model, the proposed framework can be used to explain a vast repertoire of evolutionary dynamics, including some novel forms of g…
Parity games have been broadly studied in recent years for their applications to controller synthesis and verification. In practice, partial solvers for parity games that execute in polynomial time, while incomplete, can solve most games in publicly available benchmark suites. In this paper, we combine those partial solvers with the classical recursive algorithm for parity games due to Zielonka. We also extend partial solvers to generalized parity games that are games with conjunction of parity objectives. We have implemented those algorithms and evaluated them on a large set of benchmarks pro…
Most work in game theory assumes that players are perfect reasoners and have common knowledge of all significant aspects of the game. In earlier work, we proposed a framework for representing and analyzing games with possibly unaware players, and suggested a generalization of Nash equilibrium appropriate for games with unaware players that we called generalized Nash equilibrium. Here, we use this framework to analyze other solution concepts that have been considered in the game-theory literature, with a focus on sequential equilibrium. We also provide some insight into the notion of generalize…
We study a type of generalized recursive game introduced by Castronova, Chen, and Zumbrun featuring increasing stakes, with an emphasis on continuous guts poker and $1$ v. $n$ coalitions. Our main results are to develop practical numerical algorithms with rigorous underlying theory for the approximation of optimal mutiplayer strategies, and to use these to obtain a number of interesting observations about guts. Outcomes are a striking 2-strategy optimum for $n$-player coalitions, with asymptotic advantage approximately $16\%$; convergence of Fictitious Play to symmetric Nash equilibrium; and a…
A generalized model of games is proposed, in which cooperative games and non-cooperative games are special cases. Some games that are neither cooperative nor non-cooperative can be expressed and analyzed. The model is based on relationships and supposed relationships between players. A relationship is a numerical value that denotes how one player cares for the payoffs of another player, while a supposed relationship is another numerical value that denotes a player's belief about the relationship between two players. The players choose their strategies by taking into consideration not only the …
In mean-payoff games, the objective of the protagonist is to ensure that the limit average of an infinite sequence of numeric weights is nonnegative. In energy games, the objective is to ensure that the running sum of weights is always nonnegative. Generalized mean-payoff and energy games replace individual weights by tuples, and the limit average (resp. running sum) of each coordinate must be (resp. remain) nonnegative. These games have applications in the synthesis of resource-bounded processes with multiple resources. We prove the finite-memory determinacy of generalized energy games and sh…
We solve the stochastic generalized Nash equilibrium (SGNE) problem in merely monotone games with expected value cost functions. Specifically, we present the first distributed SGNE seeking algorithm for monotone games that requires one proximal computation (e.g., one projection step) and one pseudogradient evaluation per iteration. Our main contribution is to extend the relaxed forward-backward operator splitting by Malitsky (Mathematical Programming, 2019) to the stochastic case and in turn to show almost sure convergence to a SGNE when the expected value of the pseudogradient is approximated…
In this paper, we present three distributed algorithms to solve a class of generalized Nash equilibrium (GNE) seeking problems in strongly monotone games. The first one (SD-GENO) is based on synchronous updates of the agents, while the second and the third (AD-GEED and AD-GENO) represent asynchronous solutions that are robust to communication delays. AD-GENO can be seen as a refinement of AD-GEED, since it only requires node auxiliary variables, enhancing the scalability of the algorithm. Our main contribution is to prove converge to a variational GNE of the game via an operator-theoretic appr…
We address the problem of assessing the robustness of the equilibria in uncertain, multi-agent games. Specifically, we focus on generalized Nash equilibrium problems in aggregative form subject to linear coupling constraints affected by uncertainty with a possibly unknown probability distribution. Within a data-driven context, we apply the scenario approach paradigm to provide a-posteriori feasibility certificates for the entire set of generalized Nash equilibria of the game. Then, we show that assessing the violation probability of such set merely requires to enumerate the constraints that ``…
Games on graphs provide a natural and powerful model for reactive systems. In this paper, we consider generalized reachability objectives, defined as conjunctions of reachability objectives. We first prove that deciding the winner in such games is $\PSPACE$-complete, although it is fixed-parameter tractable with the number of reachability objectives as parameter. Moreover, we consider the memory requirements for both players and give matching upper and lower bounds on the size of winning strategies. In order to allow more efficient algorithms, we consider subclasses of generalized reachability…
This paper investigates the game theory of resource-allocation situations where the "first come, first serve" heuristic creates inequitable, asymmetric benefits to the players. Specifically, this problem is formulated as a Generalized Nash Equilibrium Model where the players are arranged sequentially along a directed line graph. The goal of the model is to reduce the asymmetric benefits among the players using a policy instrument. It serves as a more realistic, alternative approach to the line-graph models considered in the cooperative game-theoretic literature. An application-oriented formula…
The AB game is a two-player game, where the codemaker has to choose a secret code and the codebreaker has to guess it in as few questions as possible. It is a variant of the famous Mastermind game, with the only difference that all pegs in both, the secret and the questions must have distinct colors. In this work, we consider the Generalized AB game, where for given arbitrary numbers $p$, $c$ with $p \le c$ the secret code consists of $p$ pegs each having one of $c$ colors and the answer consists only of a number of black and white pegs. There the number of black pegs equals the number of pegs…
We study the variant of the stable marriage problem in which the preferences of the agents are allowed to include indifferences. We present a mechanism for producing Pareto-stable matchings in stable marriage markets with indifferences that is group strategyproof for one side of the market. Our key technique involves modeling the stable marriage market as a generalized assignment game. We also show that our mechanism can be implemented efficiently. These results can be extended to the college admissions problem with indifferences.
We introduce and study the Generalized Cops and Robbers game (GCR), an N-player pursuit game in graphs. The two-player version is essentially equivalent to the classic Cops and Robbers (CR) game. The three-player version can be understood as two CR games played simultaneously on the same graph; a player can be at the same time both pursuer and evader. The same is true for four or more players. We formulate GCR as a discounted stochastic game of perfect information and prove that, for three or more players, it has at least two Nash Equilibria: one in positional deterministic strategies and anot…
We study the network replicator equation and characterize its fixed points on arbitrary graph structures for $2 \times 2$ symmetric games. We show a relationship between the asymptotic behavior of the network replicator and the existence of an independent vertex set in the graph and also show that complex behavior cannot emerge in $2 \times 2$ games. This links a property of the dynamical system with a combinatorial graph property. We contrast this by showing that ordinary rock-paper-scissors (RPS) exhibits chaos on the 3-cycle and that on general graphs with $\geq 3$ vertices the network repl…
The decision making and management of many engineering networks involves multiple parties with conflicting interests, while each party is constituted with multiple agents. Such problems can be casted as a multi-cluster game. Each cluster is treated as a self-interested player in a non-cooperative game where agents in the same cluster cooperate together to optimize the payoff function of the cluster. In a large-scale network, the information of agents in a cluster can not be available immediately for agents beyond this cluster, which raise challenges to the existing Nash equilibrium seeking alg…
Simple stochastic games are turn-based 2.5-player games with a reachability objective. The basic question asks whether one player can ensure reaching a given target with at least a given probability. A natural extension is games with a conjunction of such conditions as objective. Despite a plethora of recent results on the analysis of systems with multiple objectives, the decidability of this basic problem remains open. In this paper, we present an algorithm approximating the Pareto frontier of the achievable values to a given precision. Moreover, it is an anytime algorithm, meaning it can be …
We consider the problem of two-player zero-sum games. This problem is formulated as a min-max Markov game in the literature. The solution of this game, which is the min-max payoff, starting from a given state is called the min-max value of the state. In this work, we compute the solution of the two-player zero-sum game utilizing the technique of successive relaxation that has been successfully applied in the literature to compute a faster value iteration algorithm in the context of Markov Decision Processes. We extend the concept of successive relaxation to the setting of two-player zero-sum g…
We propose a two-layer, semi-decentralized algorithm to compute a local solution to the Stackelberg equilibrium problem in aggregative games with coupling constraints. Specifically, we focus on a single-leader, multiple-follower problem, and after equivalently recasting the Stackelberg game as a mathematical program with complementarity constraints (MPCC), we iteratively convexify a regularized version of the MPCC as inner problem, whose solution generates a sequence of feasible descent directions for the original MPCC. Thus, by pursuing a descent direction at every outer iteration, we establi…
Shared-constraint games are noncooperative $N$-player games where players are coupled through a common coupling constraint. It is known that such games admit two kinds of equilibria -- generalized Nash equilibria (GNE) and variational equilibria (VE) -- with two different economic interpretations. We consider such games in the context of resource allocation, where players move simultaneously to decide portions of the resource they can consume under a coupling constraint that the sum of the portions they demand be no more than the capacity of the resource. We clarify the worst case and best cas…
We consider Blackwell approachability, a very powerful and geometric tool in game theory, used for example to design strategies of the uninformed player in repeated games with incomplete information. We extend this theory to "generalized quitting games" , a class of repeated stochastic games in which each player may have quitting actions, such as the Big-Match. We provide three simple geometric and strongly related conditions for the weak approachability of a convex target set. The first is sufficient: it guarantees that, for any fixed horizon, a player has a strategy ensuring that the expecte…
We present a general framework for solving a large class of learning problems with non-linear functions of classification rates. This includes problems where one wishes to optimize a non-decomposable performance metric such as the F-measure or G-mean, and constrained training problems where the classifier needs to satisfy non-linear rate constraints such as predictive parity fairness, distribution divergences or churn ratios. We extend previous two-player game approaches for constrained optimization to a game between three players to decouple the classifier rates from the non-linear objective,…
This paper presents a new primal-dual method for computing an equilibrium of generalized (continuous) Nash game (referred to as generalized Nash equilibrium problem (GNEP)) where each player's feasible strategy set depends on the other players' strategies. The method is based on a new form of Lagrangian function with a quadratic approximation. First, we reformulate a GNEP as a saddle point computation problem using the new Lagrangian and establish equivalence between a saddle point of the Lagrangian and an equilibrium of the GNEP. We then propose a simple algorithm that is convergent to the sa…
In cooperative game theory, associated games allow for providing meaningful characterizations of solution concepts. Moreover, generalized values allow computing an influence or power index of each coalition in a game. In this paper, we view associated games through the lens of game maps and we define the novel Shapley-Hodge Game, briefly ``SHoGa''. We characterize SHoGa via an axiomatic approach as a generalized value, and we thoroughly discuss the consistency properties of its associated game. Furthermore, we describe the Hodge decomposition of an oriented graph representing the transitive cl…
This paper generalizes L.S. Shapley's celebrated value allocation theory on coalition games by discovering and applying a fundamental connection between stochastic path integration driven by canonical time-reversible Markov chains and Hodge-theoretic discrete Poisson's equations on general weighted graphs. More precisely, we begin by defining cooperative games on general graphs and generalize Shapley's value allocation formula for those games in terms of stochastic path integral driven by the associated canonical Markov chain. We then show the value allocation operator, one for each player def…
We present two distributed algorithms for the computation of a generalized Nash equilibrium in monotone games. The first algorithm follows from a forward-backward-forward operator splitting, while the second, which requires the pseudo-gradient mapping of the game to be cocoercive, follows from the forward-backward-half-forward operator splitting. Finally, we compare them with the distributed, preconditioned, forward-backward algorithm via numerical experiments.
We consider strongly monotone games with convex separable coupling constraints, played by dynamical agents, in a partial-decision information scenario. We start by designing continuous-time fully distributed feedback controllers, based on consensus and primal-dual gradient dynamics, to seek a generalized Nash equilibrium in networks of single-integrator agents. Our first solution adopts a fixed gain, whose choice requires the knowledge of some global parameters of the game. To relax this requirement, we conceive a controller that can be tuned in a completely decentralized fashion, thanks to th…
One of the main results shown through Roughgarden's notions of smooth games and robust price of anarchy is that, for any sum-bounded utilitarian social function, the worst-case price of anarchy of coarse correlated equilibria coincides with that of pure Nash equilibria in the class of weighted congestion games with non-negative and non-decreasing latency functions and that such a value can always be derived through the, so called, smoothness argument. We significantly extend this result by proving that, for a variety of (even non-sum-bounded) utilitarian and egalitarian social functions and fo…
We consider for the first time a stochastic generalized Nash equilibrium problem, i.e., with expected-value cost functions and joint feasibility constraints, under partial-decision information, meaning that the agents communicate only with some trusted neighbours. We propose several distributed algorithms for network games and aggregative games that we show being special instances of a preconditioned forward-backward splitting method. We prove that the algorithms converge to a generalized Nash equilibrium when the forward operator is restricted cocoercive by using the stochastic approximation …
We provide several tests to determine whether a game is a potential game or whether it is a zero-sum equivalent game---a game which is strategically equivalent to a zero-sum game in the same way that a potential game is strategically equivalent to a common interest game. We present a unified framework applicable for both potential and zero-sum equivalent games by deriving a simple but useful characterization of these games. This allows us to re-derive known criteria for potential games, as well as obtain several new criteria. In particular, we prove (1) new integral tests for potential games a…
We propose a parametric family of measures of fairness in allocations of TU-cooperative games. Their definition is based on generalized Renyi Entropy, is related to the Cowell-Kuga generalized entropy indices in welfare economics, and aims to parallel the spirit of the notion of price of anarchy in the case of convex TU-cooperative games. Since computing these indices is NP-complete in general, we first upper bound the performance of a "reverse greedy" algorithm for approximately computing worst-case fairness. The result provides a general additive error guarantee in terms of two (problem depe…
In this paper, some new criteria for detecting whether a finite game is potential are proposed by solving potential equations. The verification equations with the minimal number for checking a potential game are obtained for the first time. Some connections between the potential equations and the existing characterizations of potential games are established. It is revealed that a finite game is potential if and only if its every bi-matrix sub-game is potential.
We study new classes of games, called zero-sum equivalent games and zero-sum equivalent potential games, and prove decomposition theorems involving these classes of games. We say that two games are "strategically equivalent" if, for every player, the payoff differences between two strategies (holding other players' strategies fixed) are identical. A zero-sum equivalent game is a game that is strategically equivalent to a zero-sum game; a zero-sum equivalent potential game is a zero-sum equivalent game that is strategically equivalent to a common interest game. We also call a game "normalized" …
An algorithm is proposed to verify whether a finite game is a weighted potential game (WPG) without pre-knowledge on its weights. Then the algorithm is also applied to find the closest WPG for a given finite game. The concept and criterion of near weighted potential games (NWPGs) are given, indicating the evolutionary equivalence between a finite game and its closest WPG. Based on this, a method is proposed for replacing PGs by NWPGs in game theoretic control, which improved the applicability of PG-based optimization.
Institute of Information Management, National Chiao Tung University（台湾交通大学信息管理研究所）；Institute of Information Science, Academia Sinica（中央研究院信息科学研究所）
Different types of dynamics have been studied in repeated game play, and one of them which has received much attention recently consists of those based on "no-regret" algorithms from the area of machine learning. It is known that dynamics based on generic no-regret algorithms may not converge to Nash equilibria in general, but to a larger set of outcomes, namely coarse correlated equilibria. Moreover, convergence results based on generic no-regret algorithms typically use a weaker notion of convergence: the convergence of the average plays instead of the actual plays. Some work has been done s…
